Results of magnetization and neutron diffraction measurements of the manganese vanadate system Mn$_5$(VO$_4$)$_2$(OH)$_4$ are reported. The crystal structure of this compound contains triangular [Mn$_3$O$_{13}$] building blocks that produce two-dimensional Mn$^{2+}$ magnetic networks with striped triangular topologies. The Mn sheets are connected through the nonmagnetic vanadate tetrahedra extending along the $a$-axis. Magnetization measurements performed on single crystals reveal the onset of a long-range antiferromagnetic order below approximately 45 K. The magnetic structure is Néel-type with nearest-neighbor Mn atoms coupled via three or four antiferromagnetic bonds. The magnetic moments are confined within the layers and are oriented parallel to the $b$ direction. The magnitudes of ordered moments are reduced, presumably by geometrical frustration and the low-dimensionality of the lattice structure.